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    "3.8 多层感知机\n",
    "我们已经介绍了包括线性回归和softmax回归在内的单层神经网络。然而深度学习主要关注多层模型。在本节中，我们将以多层感知机（multilayer perceptron，MLP）为例，介绍多层神经网络的概念。"
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    "3.8.2 激活函数\n",
    "上述问题的根源在于全连接层只是对数据做仿射变换（affine transformation），而多个仿射变换的叠加仍然是一个仿射变换。解决问题的一个方法是引入非线性变换，例如对隐藏变量使用按元素运算的非线性函数进行变换，然后再作为下一个全连接层的输入。这个非线性函数被称为激活函数（activation function）。下面我们介绍几个常用的激活函数。"
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    "3.8.2.1 ReLU函数\n",
    "ReLU（rectified linear unit）函数提供了一个很简单的非线性变换。给定元素xx，该函数定义为\n",
    "\n",
    "ReLU(x)=max(x,0).ReLU(x)=max(x,0).\n",
    "可以看出，ReLU函数只保留正数元素，并将负数元素清零。为了直观地观察这一非线性变换，我们先定义一个绘图函数xyplot。"
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  {
   "cell_type": "code",
   "execution_count": 2,
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   "source": [
    "%matplotlib inline\n",
    "import tensorflow as tf\n",
    "from matplotlib import pyplot as plt\n",
    "import numpy as np\n",
    "import random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
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   "source": [
    "def use_svg_display():\n",
    "    # 用矢量图显示\n",
    "    %config InlineBackend.figure_format = 'svg'\n",
    "\n",
    "def set_figsize(figsize=(3.5, 2.5)):\n",
    "    use_svg_display()\n",
    "    # 设置图的尺寸\n",
    "    plt.rcParams['figure.figsize'] = figsize\n",
    "\n",
    "def xyplot(x_vals, y_vals, name):\n",
    "    set_figsize(figsize=(5, 2.5))\n",
    "    plt.plot(x_vals.numpy(), y_vals.numpy())\n",
    "    plt.xlabel('x')\n",
    "    plt.ylabel(name + '(x)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们接下来通过Tensor提供的relu函数来绘制ReLU函数。可以看到，该激活函数是一个两段线性函数。"
   ]
  },
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   "cell_type": "code",
   "execution_count": 32,
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   "source": [
    "x = tf.Variable(tf.range(-8,8,0.1),dtype=tf.float32)\n",
    "y = tf.nn.relu(x)\n",
    "xyplot(x, y, 'relu')"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "显然，当输入为负数时，ReLU函数的导数为0；当输入为正数时，ReLU函数的导数为1。尽管输入为0时ReLU函数不可导，但是我们可以取此处的导数为0。下面绘制ReLU函数的导数。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "with tf.GradientTape() as t:\n",
    "    t.watch(x)\n",
    "    y=y = tf.nn.relu(x)\n",
    "dy_dx = t.gradient(y, x)\n",
    "xyplot(x, dy_dx, 'grad of relu')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3.8.2.2 sigmoid函数\n",
    "sigmoid函数可以将元素的值变换到0和1之间："
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "sigmoid函数在早期的神经网络中较为普遍，但它目前逐渐被更简单的ReLU函数取代。在后面“循环神经网络”一章中我们会介绍如何利用它值域在0到1之间这一特性来控制信息在神经网络中的流动。下面绘制了sigmoid函数。当输入接近0时，sigmoid函数接近线性变换。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
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   "source": [
    "y = tf.nn.sigmoid(x)\n",
    "xyplot(x, y, 'sigmoid')"
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  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "ename": "SyntaxError",
     "evalue": "invalid character in identifier (<ipython-input-35-104bed255570>, line 1)",
     "output_type": "error",
     "traceback": [
      "\u001b[1;36m  File \u001b[1;32m\"<ipython-input-35-104bed255570>\"\u001b[1;36m, line \u001b[1;32m1\u001b[0m\n\u001b[1;33m    据链式法则，sigmoid函数的导数\u001b[0m\n\u001b[1;37m                     ^\u001b[0m\n\u001b[1;31mSyntaxError\u001b[0m\u001b[1;31m:\u001b[0m invalid character in identifier\n"
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    }
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   "source": [
    "据链式法则，sigmoid函数的导数\n",
    "\n",
    "sigmoid′(x)=sigmoid(x)(1−sigmoid(x)).sigmoid \n",
    "′\n",
    " (x)=sigmoid(x)(1−sigmoid(x)).\n",
    "下面绘制了sigmoid函数的导数。当输入为0时，sigmoid函数的导数达到最大值0.25；当输入越偏离0时，sigmoid函数的导数越接近0。"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "with tf.GradientTape() as t:\n",
    "    t.watch(x)\n",
    "    y=y = tf.nn.sigmoid(x)\n",
    "dy_dx = t.gradient(y, x)\n",
    "xyplot(x, dy_dx, 'grad of sigmoid')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3.8.2.3 tanh函数\n",
    "tanh（双曲正切）函数可以将元素的值变换到-1和1之间：\n",
    " .\n",
    "我们接着绘制tanh函数。当输入接近0时，tanh函数接近线性变换。虽然该函数的形状和sigmoid函数的形状很像，但tanh函数在坐标系的原点上对称。"
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    "xyplot(x, y, 'tanh')"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "依据链式法则，tanh函数的导数\n",
    "\n",
    "tanh′(x)=1−tanh2(x).\n",
    " \n",
    "下面绘制了tanh函数的导数。当输入为0时，tanh函数的导数达到最大值1；当输入越偏离0时，tanh函数的导数越接近0。"
   ]
  },
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   "cell_type": "code",
   "execution_count": 38,
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     },
     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "with tf.GradientTape() as t:\n",
    "    t.watch(x)\n",
    "    y=y = tf.nn.tanh(x)\n",
    "dy_dx = t.gradient(y, x)\n",
    "xyplot(x, dy_dx, 'grad of tanh')"
   ]
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    "3.8.3 多层感知机\n",
    "多层感知机就是含有至少一个隐藏层的由全连接层组成的神经网络，且每个隐藏层的输出通过激活函数进行变换。多层感知机的层数和各隐藏层中隐藏单元个数都是超参数。以单隐藏层为例并沿用本节之前定义的符号，多层感知机按以下方式计算输出：\n",
    "\n",
    "H=ϕ(XWh+bh), O=HWo+bo,\n",
    "其中ϕ表示激活函数。在分类问题中，我们可以对输出O做softmax运算，并使用softmax回归中的交叉熵损失函数。 在回归问题中，我们将输出层的输出个数设为1，并将输出O直接提供给线性回归中使用的平方损失函数。"
   ]
  },
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    "小结\n",
    "多层感知机在输出层与输入层之间加入了一个或多个全连接隐藏层，并通过激活函数对隐藏层输出进行变换。\n",
    "常用的激活函数包括ReLU函数、sigmoid函数和tanh函数。"
   ]
  }
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